S. Rao Jammalamadaka

Bayes prediction in regressions with elliptical errors

Abstract In this paper the prediction problem is considered for linear regression models with elliptical errors when the Bayes prior is non-informative. We show that the Bayes prediction density under the elliptical errors assumption is exactly the same as that obtained with normally distributed errors. Thus, assuming that the errors have a normal distribution, when the true distribution is elliptical, will not lead to incorrect predictive inferences if the error variance structure is correctly specified. This extends the results of Zellner (1976). Finally, based on Monte Carlo numerical integration procedures, computations are provided in a model with multiplicative heteroscedasticity.

Bayes prediction in the linear model with spherically symmetric errors

Abstract This paper is concerned with Bayes prediction in a linear regression model when the density of the observations is given by f(y|β,τ 2 )=∫ z>0 (2π) 7minus; n 2 τ 2 n 2 {ψ(z) −2 } n 2 exp(− τ 2 2 ψ(z) −2 ||Y−Xβ|| 2 )dG(z) , where y ϵ R n , β ϵ R k , π 2 > 0, Z is a positive random variable with distribution function G , ψ (.) is a positive function, and ||·|| denotes the Euclidean norm. We show that when prior information is objective or in the conjugate family, the Bayes prediction density is the same as that when the density of the observations is normal, for any Z .

Bahadur efficiencies of spacings tests for goodness of fit

This paper is concerned with the exact Bahadur efficiencies of spacings statistics. For a general class of statistics based on a fixed number of spacings, the explicit forms of the exact slopes are derived, and it is shown that the sum of the logarithms of spacings is optimal in this class. Some results are extended to the case where the number of spacings increase with the sample size to infinity.

Another Look at Some Results on the Recursive Estimation in the General Linear Model

Abstract Written mainly for its pedagogical interest, this note deals with the computational formulas for the recursive updating of weighted least squares parameter estimates and the residual sum of squares in the general linear model under the assumption that the errors have a multivariate normal distribution. This approach simplifies considerably the derivations of Haslett (1985).

Asymptotic comparison of three tests for goodness of fit

Abstract The so-called Greenwood statistic, based on the sum of squares of the sample spacings, is known to be locally most powerful (LMP) among all tests based symmetrically on the sample spacings. On the other hand, the χ2 criterion with the number of cells equal to the number of observations, is also known to be LMP among tests based symmetrically on the observed frequencies. While the latter compares the observed and expected frequencies holding the expected number in each cell to one, the former compares the expected and observed cell-lengths holding the observed number in each cell to one. We compare here these two test statistics with still another spacings test, Σ Di log Di, on the basis of their asymptotic relative efficiency and conclude that the Greenwood statistic is superior.

On Edgeworth Expansions in Generalized Urn Models

The random vector of frequencies in a generalized urn model can be viewed as conditionally independent random variables, given their sum. Such a representation is exploited here to derive Edgeworth expansions for a “sum of functions of such frequencies,” which are also called “decomposable statistics.” Applying these results to urn models such as with- and without-replacement sampling schemes as well as the multicolor Pólya–Egenberger model, new results are obtained for the chi-square statistic, for the sample sum in a without-replacement scheme, and for the so-called Dixon statistic that is useful in comparing two samples.

Bayes and empirical bayes estimation of the probability that z > x + y

Let X, Y and Z be independent random variables with common unknown distribution F. Using the Dirichlet process prior for F and squared erro loss function, the Bayes and empirical Bayes estimators of the parameters λ(F). the probability that Z > X + Y, are derived. The limiting Bayes estimator of λ(F) under some conditions on the parameter of the process is shown to be asymptotically normal. The aysmptotic optimality of the empirical Bayes estimator of λ(F) is established. When X, Y and Z have support on the positive real line, these results are derived for randomly right censored data. This problem relates to testing whether than used discussed by Hollander and Proshcan (1972) and Chen, Hollander and Langberg (1983).

A test of goodness-of-fit based on extreme spacings with some efficiency comparisons

Tests for the goodness-of-fit problem based on sample spacings, i.e., observed distances between successive order statistics, have been used in the literature. We propose a new test based on the number of “small” and “large” spacings. The asymptotic theory under close alternative sequences is also given thus enabling one to calculate the asymptotic relative efficiencies of such tests. A comparison of the new test and other spacings tests is given.

On an interval splitting problem

Let X1, X2,..., be i.i.d. random variables, which are uniformly distributed on [0,1]. Further let I1(0) = [0, 1] and let Ik(n) denote the kth largest interval generated by the points 0, X1, X2,..., Xn-1, 1 (or equivalently, the interval corresponding to the kth largest spacing at the nth stage). This note studies the question for which classes of sequences k = k(n), will the interval Ik(n)(n) be hit (a.s.) only finitely often, as well as infinitely often.

Bayes and empirical bayes estimation of survival function under progressive censoring

The purpose of this note is to derive the Bayes and the empirical Bayes estimators of an unknown survival function F under progressively censored data with respect to the squared error loss function and a Dirichlet process prior using the fact that the posterior distribution of F given the data is a mixture of Dirichlet processes, and the assumption that the survival and the censor in0- distributions are continuous.